The classification of links up to link-homotopy

Habegger, Nathan; Lin, Xiao-Song

doi:10.1090/S0894-0347-1990-1026062-0

The classification of links up to link-homotopy

Authors: Nathan Habegger and Xiao-Song Lin
Journal: J. Amer. Math. Soc. 3 (1990), 389-419
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0894-0347-1990-1026062-0
MathSciNet review: 1026062
Full-text PDF Free Access

E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101–126. MR 19087, DOI https://doi.org/10.2307/1969218
Gilbert Baumslag, Lecture notes on nilpotent groups, Regional Conference Series in Mathematics, No. 2, American Mathematical Society, Providence, R.I., 1971. MR 0283082
Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR 0375281

J. Birman and W. Menasco, (1) Studying links via closed braids, (2) Studying links via closed braids II: Classifying links which are closed $3$-braids, (3) Closed braid representatives of split links and composite links, (4) Closed braid representatives of the unlink, preprints, Columbia Univ., 1988.

Tim D. Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products, Mem. Amer. Math. Soc. 84 (1990), no. 427, x+73. MR 1042041, DOI https://doi.org/10.1090/memo/0427

J.-Y. le Dimet, Cobordisme d’enlacements de disques, Mém. Soc. Math. France, no. 32, Supplément Bull. Soc. Math. France 116 (1988).

R. Fenn (ed.), Low dimensional topology, London Math. Soc. Lecture Note Ser., vol. 95, Cambridge Univ. Press, London, 1985.

Deborah Louise Goldsmith, Homotopy of braids—in answer to a question of E. Artin, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Springer, Berlin, 1974, pp. 91–96. Lecture Notes in Math., Vol. 375. MR 0356021
Deborah L. Goldsmith, Concordance implies homotopy for classical links in $M^{3}$, Comment. Math. Helv. 54 (1979), no. 3, 347–355. MR 543335, DOI https://doi.org/10.1007/BF02566279
Charles H. Giffen, Link concordance implies link homotopy, Math. Scand. 45 (1979), no. 2, 243–254. MR 580602, DOI https://doi.org/10.7146/math.scand.a-11839

N. Habegger and X.-S. Lin, On Milnor’s $\overline \mu$-invariants and concordance classification of links, (in preparation).

Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI https://doi.org/10.1090/S0273-0979-1985-15304-2
J. P. Levine, Surgery on links and the $\overline \mu $-invariants, Topology 26 (1987), no. 1, 45–61. MR 880507, DOI https://doi.org/10.1016/0040-9383%2887%2990020-6
J. P. Levine, An approach to homotopy classification of links, Trans. Amer. Math. Soc. 306 (1988), no. 1, 361–387. MR 927695, DOI https://doi.org/10.1090/S0002-9947-1988-0927695-7

X.-S. Lin, Artin-type representation theorems and Milnor’s $\overline \mu$-invariants, Ph.D. thesis, Univ. of California, San Diego, 1988.

W. S. Massey, Higher order linking numbers, Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle, Chicago, Ill., 1968) Univ. of Illinois at Chicago Circle, Chicago, Ill., 1969, pp. 174–205. MR 0254832
John Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177–195. MR 71020, DOI https://doi.org/10.2307/1969685
John Milnor, Isotopy of links. Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 280–306. MR 0092150

W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory, Pure and Appl. Math., vol. XIII, Interscience, NY, 1966.

Kent E. Orr, Homotopy invariants of links, Invent. Math. 95 (1989), no. 2, 379–394. MR 974908, DOI https://doi.org/10.1007/BF01393902
Richard Porter, Milnor’s $\bar \mu $-invariants and Massey products, Trans. Amer. Math. Soc. 257 (1980), no. 1, 39–71. MR 549154, DOI https://doi.org/10.1090/S0002-9947-1980-0549154-9
John Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170–181. MR 175956, DOI https://doi.org/10.1016/0021-8693%2865%2990017-7
V. G. Turaev, The Milnor invariants and Massey products, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 189–203, 209–210 (Russian, with English summary). Studies in topology, II. MR 0451251

E. Witten, Quantum field theory and the Jones polynomial, preprint, Institute for Advanced Study, Princeton, NJ, 1988.